Signals and Systems – Relation between Convolution and Correlation

Convolution

The convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.

Mathematically, the convolution of two signals is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$

Correlation

The correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two types viz. cross-correlation and autocorrelation.

The cross correlation between two complex signals 𝑥₁(𝑡) and 𝑥₂(𝑡) is given by,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x^{\ast }_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x^{*}_{2}\left ( t \right )dt}$$

If 𝑥₁(𝑡) and 𝑥₂(𝑡) are real signals, then,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x_{2}\left ( t \right )dt}$$

Relation between Convolution and Correlation

The convolution and correlation are closely related. In order to obtain the crosscorrelation of two real signals 𝑥₁(𝑡) and 𝑥₂(𝑡), we multiply the signal 𝑥₁(𝑡) with function 𝑥₂(𝑡) displaced by τ units. Then, the area under the product curve is the cross correlation between the signals 𝑥₁(𝑡) and 𝑥₂(𝑡) at 𝑡 = 𝜏.

On the other hand, the convolution of signals 𝑥₁(𝑡) and 𝑥₂(𝑡) at 𝑡 = 𝜏 is obtained by folding the function 𝑥₂(𝑡) backward about the vertical axis at the origin [i.e., 𝑥₂(−𝑡)] and then multiplied. The area under the product curve is the convolution of the signals 𝑥₁(𝑡) and 𝑥₂(𝑡) at 𝑡 = 𝜏.

Therefore, it follows that the correlation of signals 𝑥₁(𝑡) and 𝑥₂(𝑡) is the same as the convolution of signals 𝑥₁(𝑡) and 𝑥₂(−𝑡).

Analytical Explanation

The resemblance between the convolution and the correlation can be proved analytically as follows −

The convolution of two signals 𝑥₁(𝑡) and 𝑥₂(−𝑡) is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( \tau-t \right )d\tau\: \: \: \cdot \cdot \cdot \left ( 1 \right )}$$

By replacing the variable 𝜏 by another variable p in the integral of Eqn. (1), we get,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-t \right )dp\: \: \: \cdot \cdot \cdot \left ( 2 \right )}$$

Now, replacing the variable 𝑡 by 𝜏 in Eqn. (2), we have,

$$\mathrm{x_{1}\left ( \tau \right )\ast x_{2}\left ( -\tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-\tau \right )dp=R_{12}\left ( \tau \right )}$$

Therefore, the relation between correlation and convolution of two signals is given by,

$$\mathrm{R_{12}\left ( \tau \right )=x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )|_{t=\tau }}$$

Similarly,

$$\mathrm{R_{21}\left ( \tau \right )=x_{2}\left ( t \right )\ast x_{1}\left ( -t \right )|_{t=\tau }}$$

Hence, it proves that the correlation of signals 𝑥₁(𝑡) and 𝑥₂(𝑡) is equivalent to the convolution of signals 𝑥₁(𝑡) and 𝑥₂(−𝑡).

Therefore, all the techniques used to evaluate the convolution of two signals can also be applied to find the correlation of the signals directly. Similarly, all the results obtained for the convolution are applicable to the correlation.

Note – If one of the signals is an even signal, let the signal 𝑥₂(𝑡) is an even signal [i.e. 𝑥₂(𝑡) = 𝑥₂(−𝑡)]. Then, the cross-correlation and the convolution of the two signals are equivalent.